Lemma 5.23.12. The inverse limit of a directed inverse system of finite sober topological spaces is a spectral topological space.

Proof. Let $I$ be a directed set. Let $X_ i$ be an inverse system of finite sober spaces over $I$. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ which exists by Lemma 5.14.1. As a set $X = \mathop{\mathrm{lim}}\nolimits X_ i$. Denote $p_ i : X \to X_ i$ the projection. Because $I$ is directed we may apply Lemma 5.14.2. A basis for the topology is given by the opens $p_ i^{-1}(U_ i)$ for $U_ i \subset X_ i$ open. Since an open covering of $p_ i^{-1}(U_ i)$ is in particular an open covering in the profinite topology, we conclude that $p_ i^{-1}(U_ i)$ is quasi-compact. Given $U_ i \subset X_ i$ and $U_ j \subset X_ j$, then $p_ i^{-1}(U_ i) \cap p_ j^{-1}(U_ j) = p_ k^{-1}(U_ k)$ for some $k \geq i, j$ and open $U_ k \subset X_ k$. Finally, if $Z \subset X$ is irreducible and closed, then $p_ i(Z) \subset X_ i$ is irreducible and therefore has a unique generic point $\xi _ i$ (because $X_ i$ is a finite sober topological space). Then $\xi = \mathop{\mathrm{lim}}\nolimits \xi _ i$ is a generic point of $Z$ (it is a point of $Z$ as $Z$ is closed). This finishes the proof. $\square$

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